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February 24, 2017

Responsibility

Many years ago, when I was an assistant professor at Princeton, there was a cocktail party at Curt Callan’s house to mark the beginning of the semester. There, I found myself in the kitchen, chatting with Sacha Polyakov. I asked him what he was going to be teaching that semester, and he replied that he was very nervous because — for the first time in his life — he would be teaching an undergraduate course. After my initial surprise that he had gotten this far in life without ever having taught an undergraduate course, I asked which course it was. He said it was the advanced undergraduate Mechanics course (chaos, etc.) and we agreed that would be a fun subject to teach. We chatted some more, and then he said that, on reflection, he probably shouldn’t be quite so worried. After all, it wasn’t as if he was going to teach Quantum Field Theory, “That’s a subject I’d feel responsible for.”

This remark stuck with me, but it never seemed quite so poignant until this semester, when I find myself teaching the undergraduate particle physics course.

December 4, 2016

MathML Update

For a while now, Frédéric Wang has been urging me to enable native MathML rendering for Safari. He and his colleagues have made many improvements to Webkit’s MathML support. But there were at least two show-stopper bugs that prevented me from flipping the switch.

June 14, 2016

Coriolis

I really like the science fiction TV series The Expanse. In addition to a good plot and a convincing vision of human society two centuries hence, it depicts, as Phil Plait observes, a lot of good science in a matter-of-fact, almost off-hand fashion. But one scene (really, just a few dialogue-free seconds in a longer scene) has been bothering me. In it, Miller, the hard-boiled detective living on Ceres, pours himself a drink. And we see — as the whiskey slowly pours from the bottle into the glass — that the artificial gravity at the lower levels (where the poor people live) is significantly weaker than near the surface (where the rich live) and that there’s a significant Coriolis effect. Unfortunately, the effect depicted is 3 orders-of-magnitude too big.

January 10, 2016

BMiSsed

There’s a general mantra that we all repeat to ourselves: gauge transformations are not symmetries; they are redundancies of our description. There is an exception, of course: gauge transformations that don’t go to the identity at infinity aren’t redundancies; they are actual symmetries.

Strominger, rather beautifully showed that BMS supertranslations (or, more precisely, a certain diagonal subgroup of BMS+\text{BMS}^+ (which act as supertranslations on ℐ+\mathcal{I}^+) and BMS−\text{BMS}^- (which act as supertranslations on ℐ−\mathcal{I}^-) are symmetries of the gravitational S-matrix. The corresponding conservation laws are equivalent to Weinberg’s Soft-Graviton Theorem. Similarly, in electromagnetism, the U(1)U(1) gauge transformations which don’t go to the identity on ℐ±\mathcal{I}^\pm give rise to the Soft-Photon Theorem.

A while back, there was considerable brouhaha about Hawking’s claim that BMS symmetry had something to do with resolving the blackhole information paradox. Well, finally, a paper from Hawking, Perry and Strominger has arrived.

June 19, 2015

Asymptotic Safety and the Gribov Ambiguity

Recently, an old post of mine about the Asymptotic Safety program for quantizing gravity received a flurry of new comments. Inadvertently, one of the pseudonymous commenters pointed out yet another problem with the program, which deserves a post all its own.

Before launching in, I should say that

Everything I am about to say was known to Iz Singer in 1978. Though, as with the corresponding result for nonabelian gauge theory, the import seems to be largely unappreciated by physicists working on the subject.

I would like to thank Valentin Zakharevich, a very bright young grad student in our Math Department for a discussion on this subject, which clarified things greatly for me.

May 12, 2015

Action-Angle Variables

This semester, I taught the Graduate Mechanics course. As is often the case, teaching a subject leads you to rethink that you thought you understood, sometimes with surprising results.

The subject for today’s homily is Action-Angle variables.

Let (ℳ,ω)(\mathcal{M},\omega) be a 2n2n-dimensional symplectic manifold. Let us posit that ℳ\mathcal{M} had a foliation by nn-dimensional
Lagrangian tori (a torus, T⊂MT\subset M, is Lagrangian if ω|T=0\omega|_T =0). Removing a subset, S⊂ℳS\subset \mathcal{M}, of codimension codim(S)≥2codim(S)\geq 2, where the leaves are singular, we can assume that all of the leaves on ℳ′=ℳ\S\mathcal{M}'=\mathcal{M}\backslash S are
smooth tori of dimension nn.

The objective is to construct coordinates φi,Ki\varphi^i, K_i with the following properties.

The φi\varphi^i restrict to angular coordinates on the tori. In particular φi\varphi^i shifts by 2π2\pi when you go around the corresponding cycle on TT.

The KiK_i are globally-defined functions on ℳ\mathcal{M} which are constant on each torus.

The symplectic form ω=dKi∧dφi\omega= d K_i\wedge d \varphi^i.

From 1, it’s clear that it’s more convenient to work with the 1-forms dφid\varphi^i, which are single-valued (and closed, but not necessarily exact), rather than with the φi\varphi^i themselves. In 2, it’s rather important that the KiK_i are really
globally-defined. In particular, an integrable Hamiltonian is a function H(K)H(K). The KiK_i are the nn conserved quantities which make the Hamiltonian integrable.

Obviously, a given foliation is compatible with infinitely many “integrable Hamiltonians,” so the existence of a foliation is the more fundamental concept.

All of this is totally standard.

What never really occurred to me is that the standard construction of action-angle variables turns out to be very closely wedded to the particular case of a cotangent bundle,
ℳ=T*M\mathcal{M}=T^*M.

As far as I can tell, action-angle variables don’t even exist for foliations of more general symplectic manifolds, ℳ\mathcal{M}.

December 18, 2014

Smoke Signals, Morse Code or … ?

It seemed like a straightforward question. If you use Apple’s Contacts.app to store your contacts, you’ve surely noticed this behaviour: some of your contacts auto-magically sprout clickable links for Facetime video/audio chats, with no intervention on your part. I was curious enough to submit a query about it, via Apple’s Support Site:

Contacts.app seems to know whether each of my contacts has registered their email for FaceTime, even if I have NEVER tried to facetime with them (or call their cell-phone or …). How does it do this? Are all of the email addresses in my addressbook automatically uploaded to Apple’s servers? If so, how do I turn this off, as it seems to be a MASSIVE invasion of my privacy.

That was a month and a half ago (2014/11/02). Today, I received a response:

October 25, 2014

Wikipedia

Wow! After a decade, Wikipedia finally rolls out MathML rendering. Currently, only available (as an optional preference) to registered users. Hopefully, in a few more years, they’ll make it the default.

September 27, 2014

Shellshock and MacOSX

Most Linux Distros have released patches for the recently-discovered “Shellshock” bug in /bin/bash. Apple has not, despite the fact that it uses bash as the default system shell (/bin/sh).

If you are running a webserver, you are vulnerable. Even if you avoid the obvious pitfall of writing CGI scripts as shellscripts, you are still vulnerable if one of your Perl (or PHP) scripts calls out to system(). Even Phusion Passenger is vulnerable. And, yes, this vulnerability is being actively exploited on the Web.

Some of these look like harmless probes; others (like the one which tries to download and run an IRCbot on your machine) less so.

If you’re not running a webserver, the danger is less clear. There are persistent (but apparently incorrect) rumours that Apple’s DHCP client may be vulnerable. If true, then your iPhone could easily be pwned by a rogue DHCP server (running on someone’s laptop) at Starbucks.

I don’t know what to do about your iPhone, but at least you can patch your MacOSX machine yourself.

August 19, 2014

Golem V

For nearly 20 years, Golem has been the machine on my desk. It’s been my mail server, web server, file server, … ; it’s run Mathematica and TeX and compiled software for me. Of course, it hasn’t been the same physical machine all these years. Like Doctor Who, it’s gone through several reincarnations.

Alas, word came down from the Provost that all “servers” must move (physically or virtually) to the University Data Center. And, bewilderingly, the machine on my desk counted as a “server.”

February 24, 2014

Questions

My eldest turned 18 and voted in her first Primary election this week. This being Texas, she decided to register as a Republican. Which means that, soon, we will start fielding phone calls from political campaigns. So I drafted a set of questions to ask the earnest campaign workers when they call.

February 22, 2014

Lying

Sometimes, for the sake of pedagogy, it is best to suppress some of the ugly details, in order to give a clear exposition of the idea behind a particular concept one is trying to teach. But clarity isn’t achieved by outright lies. And I always find myself frustrated when our introductory courses descend to the latter.

My colleague, Sonia, is teaching the introductory “Waves” course (Phy 315) which, as you might imagine, is all about solving the equation

To “illustrate” this, in their compulsory lab accompanying the course, the students were given the task of measuring the normal modes of a thin metal bar, with free boundary conditions at each end, sinusoidally driven by an electromagnet (of adjustable frequency).

Unfortunately, this “illustration” is a complete lie. The transverse oscillations of the metal bar are governed by an equation which is not even approximately like (1); the dispersion relation looks nothing like (2); “free boundary conditions” look nothing like (3) and therefore it should not surprise you that the normal modes look nothing like (4).

Unfortunately, so inured are they to this sort of thing, that only one (out of 120!) students noticed that something was amiss in their experiment. “Hey,” he emailed Sonia, “Why is the n=1n=1 mode absent?”

February 7, 2014

Audiophilia

Humans are hard-wired to find patterns.

Even when there are none.

Explaining those patterns (at least, the ones which are real) is what science is all about. But, even there, lie pitfalls. Have you really controlled for all of the variable which might have led to the result?

December 28, 2013

The Bus Stop Problems

Young Economics whippersnapper Evan Soltas posed two problems to do with Bayesian probability:

You arrive at a bus stop in an unfamiliar part of town. Assume that buses arrive at the stop as a Poisson process, with an unknown (to you) rate, λ\lambda. You don’t know λ\lambda, but say you have a prior probability distribution for it, p0(λ)p_0(\lambda).

What’s your expected wait time, ⟨T⟩\langle T\rangle, for the next bus to arrive?

Let’s add some more information. Say that riders arrive at the bus stop via an independent Poisson process with an (unknown to you) rate, μ\mu. Whenever a bus arrives, all those waiting at the stop get on it. Thus, the number of people waiting is the number who arrived since the last bus. Say you arrive at the stop to find nn people already waiting. You wait for a time, tt, at which point there are NN other people waiting at the stop (i.e., N−nN-n arrived while you were waiting).

These questions illustrate one of my favourite points of view on Bayes Theorem, namely that it induces a flow on the (infinite-dimensional!) space of probability distributions. Understanding the nature of that flow is, I think, the key task of the subject.

Infinite dimensions are hard to get an intuition for, so one of the first tasks is to cut the problem down to a finite-dimensional one.